Peter has a good point - if you also show that all integers have prime
factorizations (easy to do) then your method DOES prove that there are
infinitely many primes because it proves that there exists an integer Q,
larger than Pn, which is not divisible by any of the primes below (or equal
to) Pn - so either this number Q is itself prime, in which case you've
found a prime which is not in your original set, or if Q is not prime, it
must be divisible by at least one prime factor which is not contained in
your original set. Since Q is not even, this prime factor is not 2. Since
your set contains all primes between 2 and Pn, inclusive of Pn, you have
shown that there is at least one prime which is not in your set but is
larger than Pn.

On Wed, Sep 6, 2017 at 3:56 PM, peter green <plugwash@p10link.net> wrote:

> On 06/09/17 17:27, Isaac M. Bavaresco wrote:
> > Dear All,
> >
> >
> > Today I woke up with a silly idea about prime numbers in my head:
> >
> > What is the proof that there are infinitely many prime numbers? One of
> > such proofs was in my mind.
> >
> > Then I Googled and found Euclid's Proof. It is much similar to mine but
> > not exactly the same.
> >
> >
> > My proof:
> >
> > Consider a finite list of consecutive prime numbers starting in 3: 3, 5=
,
> > 7, 11, ..., Pn.
> >
> > Let P be the product of all the prime numbers in the list.
> >
> > Let Q =3D P + 2. Let's prove that Q is prime:
> >
> > P + 1 is even (not prime)
> >
> > P + 3 is multiple of 3 (not prime)
> >
> > P + 5 is multiple of 5 (not prime)
> >
> > ...
> >
> > P + Pn is multiple of Pn (not prime)
> >
> >
> > So Q cannot be multiple of any of the numbers in the list,
> True
> > thus Q is prime.
> This is where you go wrong.
>
> There may be prime numbers which are larger than Pn but smaller than Q.
> Some of these could conceivably be factors of Q.
>
> For example
>
> (3x5x7x11)+2 =3D 1157 =3D 13*89
>
> So while this proves there is at least one prime that is not on your list
> (and hence proves there are infinitely many primes) it does not provide a=
n
> efficient method for actually finding large primes.
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